imap - Dropping connection (max connections exceeded)

Zimbra 8.0.5
CentOS 6.4

New install as of today. All seemed well at first, then I started experiencing significant queueing problems in the Active Queue, which had me looking at all kinds of possible queueing issues. After a while, my users started complaining about not being able to connect via IMAP, and/or losing their existing IMAP connections. This revealed a whole bunch of messages like this:

in the mailbox.log. So, I increased zimbraImapMaxConnections to 300 (from 200), and things cleared up a bit. A little while later, though, and they started complaining about IMAP again. Again, I saw these messages in the logs. So I bumped zimbraImapMaxConnections up to 500, which again cured the problem for a short while. But, before long, the users were once again complaining. At this point, I've set zimbraImapMaxConnections to 1000, and netstat shows me with 709 IMAP connections (it doesn't look like any one source IP is at fault).

How about some server specifications including number of IMAP users on the server (any virtualization)? It may just be a slow server, what about this section of the tuning wiki article and a forum search for starters?

We have about 2500 accounts on the server, of which no more than 250 are ever active at one time - some use Imap, some use Pop, some use ZCS. It's a fast, big AWS EC2 server (set at m3.2xlarge right now: 30GB, 8 virtual CPUs).